Domanda del libro di testo
Evaluate the integrals in Exercises 33–54.
∫(from ln4 to ln9)e^(x/2)dx
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.3.21
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = ln(e^(θ)/(1+e^θ))
Guida verificata passo dopo passo1
Rewrite the given function to simplify the expression inside the logarithm: \(y = \ln\left( \frac{e^{\theta}}{1 + e^{\theta}} \right)\).
Use the logarithm property \(\ln\left( \frac{a}{b} \right) = \ln(a) - \ln(b)\) to separate the function into \(y = \ln(e^{\theta}) - \ln(1 + e^{\theta})\).
Simplify \(\ln(e^{\theta})\) using the property \(\ln(e^{x}) = x\), so the function becomes \(y = \theta - \ln(1 + e^{\theta})\).
Differentiate each term with respect to \(\theta\): the derivative of \(\theta\) is 1, and for \(-\ln(1 + e^{\theta})\), apply the chain rule.
For the second term, use the chain rule: \(\frac{d}{d\theta} \left[ -\ln(1 + e^{\theta}) \right] = - \frac{1}{1 + e^{\theta}} \cdot \frac{d}{d\theta} (1 + e^{\theta}) = - \frac{e^{\theta}}{1 + e^{\theta}}\).
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Derivative of the Natural Logarithm Function
The derivative of ln(u), where u is a differentiable function of a variable, is (1/u) times the derivative of u. This rule allows us to differentiate logarithmic expressions by first identifying the inner function and then applying the chain rule.
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Derivative of the Natural Logarithmic Function
Chain Rule
The chain rule is used to differentiate composite functions. It states that the derivative of a function composed of another function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
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05:02Intro to the Chain Rule
Simplification of Exponential Expressions
Simplifying expressions involving exponentials, such as e^θ/(1 + e^θ), helps in easier differentiation. Recognizing that e^θ/(1 + e^θ) can be rewritten or simplified aids in applying derivative rules more efficiently.
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6:39Simplifying Exponential Expressions
Pratica correlata
Domanda del libro di testo
41. Cooling soup Suppose that a cup of soup cooled from 90°C to 60°C after 10 min in a room where the temperature was 20°C. Use Newton’s Law of Cooling to answer the following questions.
a. How much longer would it take the soup to cool to 35°C?
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Domanda del libro di testo
Evaluate the integrals in Exercises 39–56.
41. ∫2y dy/(y²-25)
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Evaluate the integrals in Exercises 53–76.
69. ∫dx/((2x-1)√((2x-1)²-4))
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Evaluate the integrals in Exercises 39–56.
52. ∫(from π/4 to π/2)cot(t)dt
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Domanda del libro di testo
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(θ)(sinθ + cosθ)
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